Elliptic Curve Rank Record: 30

Update, August 20, 2026

Two years after the Elkies-Klagsbrun rank-29 record, the record has moved again. An unidentified user writing as ranksunbounded submitted the following elliptic curve over the rationals:

y^2 + x y = x^3
- 201769035260418549083594900060734240952308696994802735114305555 x
+ 1151107939141058565733479426024323225135665982951300586808823640527729578307228357301072889377

Thirty independent rational points have been exhibited, proving unconditionally that rank E(Q) >= 30. The curve and its witness points are available from the Elliptic Curve Rank Leaderboard and Andrej Dujella's rank-record page.

https://elliptic-rank.icarm.cloud/curve/273
https://web.math.pmf.unizg.hr/~duje/tors/rk30.html

This is more than a numerical estimate based on a nonzero regulator. The leaderboard checks that every point lies on the curve using exact rational arithmetic and certifies their independence modulo torsion using an exact quadratic-character 2-descent calculation derived from the Cremona-Brumer method. The verification code and its mathematical explanation are public:

https://github.com/icarm/elliptic-rank/blob/main/src/verify.ts

Import and check the witnesses

The data can be imported directly into the Wolfram Language. The following checks the elementary part of the certificate: all 30 listed rational points really lie on the stated curve. Certifying independence is the substantially harder calculation performed by the leaderboard.

In[]:=

Clear[a1, a2, a3, a4, a6, ainvs, data, onCurveQ, points30];

data = Import[
"https://elliptic-rank.icarm.cloud/curve/273.json",
"RawJSON"
];

ainvs = ToExpression /@ data["ainvs"];
points30 = Map[ToExpression, data["points"], {2}];

{a1, a2, a3, a4, a6} = ainvs;

onCurveQ[{x_, y_}] :=
y^2 + a1 x y + a3 y ==
x^3 + a2 x^2 + a4 x + a6;

{Length[points30], And @@ (onCurveQ /@ points30)}

Out[]=

{30,True}

What is known exactly?

At present the rigorous result is rank at least 30, not an unconditional proof that the rank equals 30. Andrew Sutherland reports that the analytic-rank method in "Conditionally Bounding Analytic Ranks of Elliptic Curves" gives an upper bound of 31 under GRH. The curve has global root number +1, so its analytic rank is even. Consequently its rank is exactly 30 under GRH together with the Birch-Swinnerton-Dyer conjecture.

https://arxiv.org/abs/1112.1503

The recent progression: Noam Elkies found rank at least 28 in 2006;
Noam Elkies and Zev Klagsbrun found rank at least 29 in 2024 (shown above);
ranksunbounded submitted rank at least 30 in 2026.

The actual coordinates contain roughly 30 to 57 digits. A signed logarithm preserves the sign and compresses the scale. This is a transformed point cloud, not a geometric plot of the elliptic curve. Points having two integral coordinates are separated from points having at least one nonintegral coordinate.

In[]:=

xUnit=10^31;yUnit=10^46;scaledPoints=N[{#[[1]]/xUnit,#[[2]]/yUnit}]&/@points30;indexedPoints=MapIndexed[{First[#2],#1}&,scaledPoints];cross=N[xUnit/yUnit];cubic=N[xUnit^3/yUnit^2];linear=N[a4xUnit/yUnit^2];constant=N[a6/yUnit^2];upper[z_]:=(-crossz+Sqrt[(crossz)^2+4(cubicz^3+linearz+constant)])/2;lower[z_]:=(-crossz-Sqrt[(crossz)^2+4(cubicz^3+linearz+constant)])/2;curveView[range_,title_]:=Module[{shown,labeled},shown=Select[indexedPoints,range[[1,1]]<=#[[2,1]]<=range[[1,2]]&&range[[2,1]]<=#[[2,2]]<=range[[2,2]]&];labeled=Select[shown,MemberQ[{6,7,11,12},#[[1]]]&];Plot[Evaluate@{upper[z],lower[z]},{z,range[[1,1]],range[[1,2]]},AspectRatio->1,Epilog->{RGBColor[.8,.18,.08],PointSize[.018],Tooltip[Point[#[[2]]],"P"<>ToString[#[[1]]]]&/@shown,Black,Text[Style["P"<>ToString[#[[1]]],11,Bold],#[[2]],{-1.2,-1.2}]&/@labeled},Frame->True,FrameLabel->{"x / 10^31","y / 10^46"},PlotLabel->Row[{title," — ",Length[shown]," witnesses"}],PlotRange->range,PlotStyle->Black]];GraphicsRow[{curveView[{{-1.6,2.8},{-12,12}},"natural scale"],curveView[{{-2,80},{-2200,2200}},"intermediate scale"]},ImageSize->600]

Out[]=

Here, each point {x,y} is mapped to {Sign[x] Log[Abs[x]], Sign[y] Log[Abs[y]]}. This compresses the enormous coordinates while retaining their signs, so the four quadrants show the four possible sign combinations.

In[]:=

logged=Sign[#]Log[Abs[#]]&/@#&/@points30;Graphics[{Black,PointSize[.012],Point[logged]},AspectRatio->1,PlotRange->All,PlotRangePadding->Scaled[.06],ImageSize->300]

Out[]=

The points are mapped to {Abs[x],Abs[y]}, ordered by Max[Abs[x],Abs[y]], and only the 23 smallest are displayed. This prevents the seven largest-coordinate points from determining the plotting range, but it discards sign information and uses a size cutoff rather than a statistical outlier test. Due to the warping from Abs, the curve does not actually join up like this.

In[]:=

outlier=SortBy[Abs[points30],Max];Graphics[{Black,PointSize[.012],Point[Take[outlier,23]]},AspectRatio->1,PlotRange->All,PlotRangePadding->Scaled[.06],ImageSize->300]

Out[]=

Sources

Elliptic Curve Rank Leaderboard, curve 273:
https://elliptic-rank.icarm.cloud/curve/273

Machine-readable curve data:
https://elliptic-rank.icarm.cloud/curve/273.json

Andrej Dujella, Rank >= 30:
https://web.math.pmf.unizg.hr/~duje/tors/rk30.html

Andrej Dujella, History of elliptic curve rank records:
https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html

Leaderboard verification code:
https://github.com/icarm/elliptic-rank/blob/main/src/verify.ts

Jonathan Bober, Conditionally Bounding Analytic Ranks of Elliptic Curves:
https://arxiv.org/abs/1112.1503